.. _chapter_padding:
Padding and Stride
==================
In the previous example, our input had a height and width of 3 and a
convolution kernel with a height and width of 2, yielding an output with
a height and a width of 2. In general, assuming the input shape is
:math:`n_h\times n_w` and the convolution kernel window shape is
:math:`k_h\times k_w`, then the output shape will be
.. math:: (n_h-k_h+1) \times (n_w-k_w+1).
Therefore, the output shape of the convolutional layer is determined by
the shape of the input and the shape of the convolution kernel window.
In several cases we might want to incorporate particular
techniquesâ€”padding and stridesâ€”regarding the size of the output:
- In general, since kernels generally have width and height greater
than 1, that means that after applying many successive convolutions,
we will wind up with an output that is much smaller than our input.
If we start with a 240x240 pixel image, 10 layers of 5x5 convolutions
reduce the image to 200x200 pixels, slicing off 30% of the image and
with it obliterating any interesting information on the boundaries of
the original image. *Padding* handles this issue.
- In some cases, we want to reduce the resolution drastically if say we
find our original input resolution to be unweildy. *Strides* can help
in these instances.
Padding
-------
As described above, one tricky issue when applying convolutional layers
is that losing pixels on the permimeter of our image. Since we typically
use small kernels, for any given convolution, we might only lose a few
pixels, but this can add up as we apply many successive convolutional
layers. One straightforward solution to this problem is to add extra
pixels of filler around the boundary of our input image, thus increasing
the effective size of the image Typically, we set the values of the
extra pixels to 0. In the figure below, we pad a :math:`3 \times 5`
input, increasing its size to :math:`5 \times 7`. The corresponding
output then increases to a :math:`4 \times 6` matrix.
.. figure:: ../img/conv-pad.svg
Two-dimensional cross-correlation with padding. The shaded portions
are the input and kernel array elements used by the first output
element: :math:`0\times0+0\times1+0\times2+0\times3=0`.
In general, if we add a total of :math:`p_h` rows of padding (roughly
half on top and half on bottom) and a total of :math:`p_w` columns of
padding (roughly half on the left and half on the right), the output
shape will be
.. math:: (n_h-k_h+p_h+1)\times(n_w-k_w+p_w+1),
This means that the height and width of the output will increase by
:math:`p_h` and :math:`p_w` respectively.
In many cases, we will want to set :math:`p_h=k_h-1` and
:math:`p_w=k_w-1` to give the input and output the same height and
width. This will make it easier to predict the output shape of each
layer when constructing the network. Assuming that :math:`k_h` is odd
here, we will pad :math:`p_h/2` rows on both sides of the height. If
:math:`k_h` is even, one possibility is to pad
:math:`\lceil p_h/2\rceil` rows on the top of the input and
:math:`\lfloor p_h/2\rfloor` rows on the bottom. We will pad both sides
of the width in the same way.
Convolutional neural networks commonly use convolutional kernels with
odd height and width values, such as 1, 3, 5, or 7. Choosing odd kernel
sizes has the benefit that we can preserve the spatial dimensionality
while padding with the same number of rows on top and bottom, and the
same number of columns on left and right.
Moreover, this practice of using odd kernels and padding to precisely
preserve dimensionality offers a clerical benefit. For any
two-dimensional array ``X``, when the kernels size is odd and the number
of padding rows and columns on all sides are the same, producing an
output with the have the same height and width as the input, we know
that the output ``Y[i,j]`` is calculated by cross-correlation of the
input and convolution kernel with the window centered on ``X[i,j]``.
In the following example, we create a two-dimensional convolutional
layer with a height and width of 3 and apply 1 pixel of padding on all
sides. Given an input with a height and width of 8, we find that the
height and width of the output is also 8.
.. code:: python
from mxnet import nd
from mxnet.gluon import nn
# For convenience, we define a function to calculate the convolutional layer.
# This function initializes the convolutional layer weights and performs
# corresponding dimensionality elevations and reductions on the input and
# output
def comp_conv2d(conv2d, X):
conv2d.initialize()
# (1,1) indicates that the batch size and the number of channels
# (described in later chapters) are both 1
X = X.reshape((1, 1) + X.shape)
Y = conv2d(X)
# Exclude the first two dimensions that do not interest us: batch and
# channel
return Y.reshape(Y.shape[2:])
# Note that here 1 row or column is padded on either side, so a total of 2
# rows or columns are added
conv2d = nn.Conv2D(1, kernel_size=3, padding=1)
X = nd.random.uniform(shape=(8, 8))
comp_conv2d(conv2d, X).shape
.. parsed-literal::
:class: output
(8, 8)
When the height and width of the convolution kernel are different, we
can make the output and input have the same height and width by setting
different padding numbers for height and width.
.. code:: python
# Here, we use a convolution kernel with a height of 5 and a width of 3. The
# padding numbers on both sides of the height and width are 2 and 1,
# respectively
conv2d = nn.Conv2D(1, kernel_size=(5, 3), padding=(2, 1))
comp_conv2d(conv2d, X).shape
.. parsed-literal::
:class: output
(8, 8)
Stride
------
When computing the cross-correlation, we start with the convolution
window at the top-left corner of the input array, and then slide it over
all locations both down and to the right. In previous examples, we
default to sliding one pixel at a time. However, sometimes, either for
computational efficiency or because we wish to downsample, we move our
window more than one pixel at a time, skipping the intermediate
locations.
We refer to the number of rows and columns traversed per slide as the
*stride*. So far, we have used strides of 1, both for height and width.
Sometimes, we may want to use a larger stride. The figure below shows a
two-dimensional cross-correlation operation with a stride of 3
vertically and 2 horizontally. We can see that when the second element
of the first column is output, the convolution window slides down three
rows. The convolution window slides two columns to the right when the
second element of the first row is output. When the convolution window
slides two columns to the right on the input, there is no output because
the input element cannot fill the window (unless we add padding).
.. figure:: ../img/conv-stride.svg
Cross-correlation with strides of 3 and 2 for height and width
respectively. The shaded portions are the output element and the
input and core array elements used in its computation:
:math:`0\times0+0\times1+1\times2+2\times3=8`,
:math:`0\times0+6\times1+0\times2+0\times3=6`.
In general, when the stride for the height is :math:`s_h` and the stride
for the width is :math:`s_w`, the output shape is
.. math:: \lfloor(n_h-k_h+p_h+s_h)/s_h\rfloor \times \lfloor(n_w-k_w+p_w+s_w)/s_w\rfloor.
If we set :math:`p_h=k_h-1` and :math:`p_w=k_w-1`, then the output shape
will be simplified to
:math:`\lfloor(n_h+s_h-1)/s_h\rfloor \times \lfloor(n_w+s_w-1)/s_w\rfloor`.
Going a step further, if the input height and width are divisible by the
strides on the height and width, then the output shape will be
:math:`(n_h/s_h) \times (n_w/s_w)`.
Below, we set the strides on both the height and width to 2, thus
halving the input height and width.
.. code:: python
conv2d = nn.Conv2D(1, kernel_size=3, padding=1, strides=2)
comp_conv2d(conv2d, X).shape
.. parsed-literal::
:class: output
(4, 4)
Next, we will look at a slightly more complicated example.
.. code:: python
conv2d = nn.Conv2D(1, kernel_size=(3, 5), padding=(0, 1), strides=(3, 4))
comp_conv2d(conv2d, X).shape
.. parsed-literal::
:class: output
(2, 2)
For the sake of brevity, when the padding number on both sides of the
input height and width are :math:`p_h` and :math:`p_w` respectively, we
call the padding :math:`(p_h, p_w)`. Specifically, when
:math:`p_h = p_w = p`, the padding is :math:`p`. When the strides on the
height and width are :math:`s_h` and :math:`s_w`, respectively, we call
the stride :math:`(s_h, s_w)`. Specifically, when :math:`s_h = s_w = s`,
the stride is :math:`s`. By default, the padding is 0 and the stride is
1. In practice we rarely use inhomogeneous strides or padding, i.e., we
usually have :math:`p_h = p_w` and :math:`s_h = s_w`.
Summary
-------
- Padding can increase the height and width of the output. This is
often used to give the output the same height and width as the input.
- The stride can reduce the resolution of the output, for example
reducing the height and width of the output to only :math:`1/n` of
the height and width of the input (:math:`n` is an integer greater
than 1).
- Padding and stride can be used to adjust the dimensionality of the
data effectively.
Exercises
---------
1. For the last example in this section, use the shape calculation
formula to calculate the output shape to see if it is consistent with
the experimental results.
2. Try other padding and stride combinations on the experiments in this
section.
3. For audio signals, what does a stride of :math:`2` correspond to?
4. What are the computational benefits of a stride larger than
:math:`1`.
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.. |image0| image:: ../img/qr_padding-and-strides.svg